Download Article - Archive ouverte UNIGE

Article
Analysing the chromium-chromium multiple bond using
multiconfigurational quantum chemistry
BRYNDA, Marcin, GAGLIARDI, Laura, ROOS, Björn O.
Abstract
This Letter discusses the nature of the chemical bond between two chromium atoms in
different di-chromium complexes with the metal atoms in different oxidation states. Starting
with the Cr diatom, with its formally sextuple bond and oxidation number zero, we proceed to
analyse the bonding in some Cr(I)–Cr(I) XCrCrX complexes with X varying from F, to Phenyl,
and Aryl. The bond distance in these complexes varies over a large range: 1.65–1.83 Å and
we suggest explanations for these variations. A number of di-chromium complexes with bond
distances around or shorter than 1.80 Å have recently been synthesized and we study one of
these complexes, Cr2(diazadiene)2 and show how the Cr–Cr bond order is related to the
oxidation number and the ligand bonding, factors that are all involved in the determination of
the short Cr–Cr bond length: 1.80 Å. The discussion is based on the use of
multiconfigurational wave functions, which give a qualitatively correct description of the
electronic structure in these multiply bonded systems.
Reference
BRYNDA, Marcin, GAGLIARDI, Laura, ROOS, Björn O. Analysing the chromium-chromium
multiple bond using multiconfigurational quantum chemistry. Chemical Physics Letters, 2009,
vol. 471, no. 1-3, p. 1-10
DOI : 10.1016/j.cplett.2009.02.006
Available at:
http://archive-ouverte.unige.ch/unige:3530
Disclaimer: layout of this document may differ from the published version.
Chemical Physics Letters 471 (2009) 1–10
Contents lists available at ScienceDirect
Chemical Physics Letters
journal homepage: www.elsevier.com/locate/cplett
FRONTIERS ARTICLE
Analysing the chromium–chromium multiple bonds using multiconfigurational
quantum chemistry
Marcin Brynda a,*, Laura Gagliardi a, Björn O. Roos b
a
b
Département de Chimie Physique, Université de Genève, 30, q. E. Ansermet, 1211 Genève, Switzerland
Department of Theoretical Chemistry, Chemical Center, P.O. Box 124 S-221 00 Lund, Sweden
a r t i c l e
i n f o
Article history:
Received 17 December 2008
In final form 2 February 2009
Available online 7 February 2009
a b s t r a c t
This Letter discusses the nature of the chemical bond between two chromium atoms in different di-chromium complexes with the metal atoms in different oxidation states. Starting with the Cr diatom, with its
formally sextuple bond and oxidation number zero, we proceed to analyse the bonding in some Cr(I)–
Cr(I) XCrCrX complexes with X varying from F, to Phenyl, and Aryl. The bond distance in these complexes
varies over a large range: 1.65–1.83 Å and we suggest explanations for these variations. A number of dichromium complexes with bond distances around or shorter than 1.80 Å have recently been synthesized
and we study one of these complexes, Cr2(diazadiene)2 and show how the Cr–Cr bond order is related to
the oxidation number and the ligand bonding, factors that are all involved in the determination of the
short Cr–Cr bond length: 1.80 Å. The discussion is based on the use of multiconfigurational wave functions, which give a qualitatively correct description of the electronic structure in these multiply bonded
systems.
Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction
Recent advances in the field of the synthetic transition metal
complexes exhibiting multiply bonded metal–metal cores [1], have
reopened many theoretical questions that have, if not been sent
completely ad-acta, at least had seen attenuated levels of interest
for almost two decades. A major reason for this was the scarcity
of new classes of multiple bonded metal–metal compounds
appearing in the literature. In fact, not much happened on the synthetic front after the pioneering experimental work of Cotton with
the synthesis of the room-temperature stable Re2 Cl8 species with a
formal quadruple bond [2] even if a large number of other quadruply bonded di-metal compounds have been synthesized and
theoretically studied since then. The search for an even higher multiplicity of the metal–metal bond was only recently successful with
the synthesis of the first room-temperature stable RCrCrR compound with terphenyl ligands by Power’s group [3]. Following
the discovery of this first stable quintuply bonded chromium–
chromium species and publication of its crystal-structure, several
low-valent di-chromium complexes were synthesized in the last
three years [4–8]. Not surprisingly, this flurry of experimental
activity has been accompanied by a number of theoretical studies,
which were undertaken to describe the bonding paradigm in these
species [8–16]. However, the effectiveness of rapidly evolving
quantum mechanical methods in handling complicated theoretical
* Corresponding author. Fax: +41 022 379 6518.
E-mail address: [email protected] (M. Brynda).
0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2009.02.006
descriptions of metal–metal multiple bonding provides ample justification for a reexamination of many of these systems. In this review we shall try to summarize several of these results and see if
there is any common pattern which may be used to understand
the metal–metal bonding in some of these molecules. In most of
the calculations that we refer to, we have used multiconfigurational quantum chemical wave functions because they capture the
essentials of the electronic structure in a compact way through
the use of the natural orbital concept and effective bond orders.
Comparison will also be made with density functional methods because they have frequently been used to analyze the electronic
structure of these molecules.
The mother of the multiply bonded Cr2 species is the Cr diatom
itself. Despite the fact that this seemingly simple molecule was
first identified and observed more than four decades ago [17,18],
the description of its electronic structure remained a longstanding
challenge for theoretical chemistry and its bonding is still an area
of some debate. We shall therefore start with a short historical prospective of the theoretical efforts to describe the bonding in the Cr2
and then discuss its recently synthesized derivatives.
2. The chromium–chromium multiple bond – a historical
overview
The simplest molecule that exhibits a Cr–Cr bond with multiple
character is Cr2. Based on laser-induced fluorescence spectroscopic
studies its experimental bond length was determined to be 1.679 Å
[19], and its dissociation energy was calculated to be
2
M. Brynda et al. / Chemical Physics Letters 471 (2009) 1–10
D0 = 1.53 ± 0.06 eV (35.3 ± 1.4 kcal mol1) [20]. The experimental
potential energy curve is shown in Fig. 1 and the details of the
experimental data are discussed in the original reference by Casey
and Leopold [21]. As we have recently noted [9], even though this
bond is considered extremely short and has a formal bond order of
six, its relevance was largely questioned [22] because of its significantly lower dissociation energy, as compared to analogous species involving atoms from the second or third row. Depending on
the reported theoretical studies, the description of the bonding in
Cr2 has ranged from a sextuple bond [22–26], through a single
bond [27], to an antiferromagnetically coupled diatomic bearing
two chromium centers with opposite spins [22].
Before we give a detailed description of the bonding in the Cr–
Cr species, it is interesting to recall the electronic structure of Cr2
and its simplest extension – RCrCrR (where R = H, Me, F, Ph or
Ar). Each chromium in the Cr2 diatom possesses five 3d and one
4s electron and can, in principle, afford a fully sextuple bond between the metal centers. The main difference between Cr2 and
the analogous RCrCrR species lies in the fact that in the latter, a pair
of 4s electrons is used to form the M–R bonds with the ligands and
therefore only the remaining 3d electrons can be involved in the
metal–metal bonding. Then, the ten 3d electrons could result in a
fivefold interaction.
With this simple picture in hand we can now consider the
development of the chemistry of multiple bonded Cr–Cr species
in general. Modern synthetic work in multiple bonded transition
metal complexes was initiated by Cotton’s quadruply bonded complexes of rhenium [2] and following this discovery, numerous
other species with multiply bonded metal–metal cores were synthesized. A large number of chromium complexes exhibiting quadruple Cr–Cr bonds were obtained at that time. Maybe the most
famous is the Cotton’s Cr2(2-MeO–5-MeC6H3)4 [28] with two Cr(II)
having an experimental Cr–Cr bond length of 1.828 Å. This remained for almost 30 years the shortest metal–metal bond known
in an isolable compound, until it was very recently superseded by
the quadruply bonded [{(Me3Si)2NC(NCy)2}2CrMe]2 complex with
a very short Cr–Cr bond of 1.773 Å [29]. It should be mentioned,
however, that the iconic quadruply bonded chromium acetate
Cr2{OC(O)Me}4 of Peligot [30,31] had been known since the
1840’s, although the nature of the Cr–Cr bond was not recognized
until much later.
The break-through discovery, which reinvigorated the concept
of multiple bonding in transition metal chemistry, took place in
2005 with the report of the first Cr–Cr complex Ar0 CrCrAr0 with a
Fig. 1. The experimental potential energy curve for Cr2 [21].
fivefold Cr–Cr interaction [3]. The Ar0 CrCrAr0 (Ar0 = C6H3-2,6(C6H32,6-Pri2)2) complex reported by Power’s group exhibits a planar
trans-bent structure with a very short CrCr bond (1.835 Å) and
a CrCrC angle of 102.8°. As in most of the complexes where terphenyl ligands are used as kinetically stabilizing agents, there is
also a relatively short (2.294 Å) secondary CrC interaction involving an ipso-carbon of one of the flanking aryl rings. Following this
experimental work, subsequent synthetic efforts resulted in several other low-valent chromium–chromium species with either
modified terphenyl ligands (including electron donating/withdrawing substituents), or with new ligands containing nitrogen.
By the substitution of the central phenyl ring of the terphenyl ligand with OMe, SiMe3 or F, three new variants of the Ar0 CrCrAr0
were obtained [8] (see Table 1). However, the synthesized structures were strikingly similar to the prototypic Ar0 CrCrAr0 and only
very small effects on the CrCr bond were noticeable. The most
important features of these analogous of Ar0 CrCrAr0 are again the
trans-bent solid-state structures with very short CrCr distances
that lie in the narrow range of 1.808–1.835 Å. The differences in
the CrCr distances could not be correlated with the electronwithdrawing or -donating power of the para substituents and
seemed rather to come from crystal-packing effects. Thus, the
Cr–Cr bond was shown to be insensitive to the electronic nature
of substituent at the para position of the central aryl ring.
However, very recent reports provide evidence that the chemistry of Cr–Cr complexes is not limited to the use of the bulky terphenyl moieties, but can easily be extended to other types of
supporting ligands. In particular, chemical entities containing
nitrogen are an emerging ligand class that can stabilize multiple
bonded metal–metal species. For example, two derivatives of diimines ligands where each chromium atom interacts with two
nearby nitrogens have recently been described. Kreisel et al. synthesized a di-chromium compound (l-g2-HLiPr)2Cr2, HLiPr = N,N0 bis(2,6-diisopropylphenyl)-1,4-diazadiene, where the Cr–Cr core
is coordinated by two diazadienes [5]. In this compound, the geometry around each Cr atom is trigonal planar, with each metal coordinated to two N atoms from two bridging diazadiene ligands, as
well as by the neighboring Cr atom. The interesting feature of this
complex is the very short CrCr distance of 1.803 Å (shorter by
0.03 Å than the previously reported Ar0 CrCrAr0 ), making it one
of the shortest metalmetal distances reported to date. It is also
important to mention that due to the particular nature of the
N2CrCrN2 core (close to planar arrangement around Cr2) in this
complex, no secondary interactions of Cr with other parts of the ligands are involved. A similar compound, L2Cr2, L = [6-(2,4,6-triisopropylphenyl)pyridin-2-yl](2,4,6-trimethylphenyl)amide, with a
slightly different ligand containing two unequivalent nitrogens
(one pyridino and one imino) was recently reported by Noor
et al. [6]. This compound is characterized by an exceptionally short
Cr–Cr distance of 1.749 Å and a rather short Cr–Namido bond length
(1.998 Å). Unlike in the previous compound, two different types of
nitrogens are involved in the bonding with the chromium core.
In addition, a further report shows that the fivefold Cr–Cr bond
does not necessarily imply planar (or close to planar) structures as
in the case of the RCrCrR or RN2CrCrN2R species, but can also be
stabilized by somehow unexpected arrangement of the ligands.
Tsai et al. have synthesized a paramagnetic molecule containing
three diimine ligands forming a cage (lantern) that hosts a Cr–Cr
core with a Cr–Cr interaction of 1.817 Å [7]. Reduction of this molecule afforded its diamagnetic congener, which is a stable entity
and exhibits the shortest Cr–Cr bond known to date of (1.740 Å)
for anything other than the Cr diatom itself.
Last but not least in the saga of the low-valent chromium–chromium complexes, four new molecules with Cr–Cr cores using modified bis(amidinate) ligands with short N–N distances were
reported, while we were submitting this review by Hsu et al. These
3
M. Brynda et al. / Chemical Physics Letters 471 (2009) 1–10
Table 1
Recently synthesized low-valent Cr–Cr species with a fivefold interaction between the chromium atoms. Experimental CrCr distances are reported from the available X-ray
structures. BO DFT – Bond order represented by Wiberg bond indices from DFT calculations; EBO – Effective Bond Order from CASPT2 calculations as defined in paragraph 4.
CrCr species
CrCr distance (Å)
BO DFT
EBO CASPT2
Cr2
PhCrCrPha Linear
PhCrCrPha Trans-bent
FCrCrFa
Ar0 CrCrAr0 Ar0 = C6H3-2,6(C6H3-2,6-Pri2)2) [3]
Ar*CrCrAr* Ar* = 4-SiMe3-Ar0 [8]
Ar¥CrCrAr¥ Ar¥ 4-OMe-Ar0 [8]
Ar§CrCrAr§ Ar§ 4-F-Ar0 [8]
[Cr2-(ArXylNC(H)NArXyl)3] [7]
(ArXyl=2,6-C6H3-(CH3)2)
[Cr2-(ArXylNC(H)NArXyl)3] [7]
(ArXyl=2,6-C6H3-(CH3)2)
(l-g2-HLiPr)2Cr2, HLiPr = N,N0 -bis(2,6-diisopropylphenyl)-1,4-diazadiene [6]
1.679
1.687b
1.750b
1.650b
1.835
1.808
1.816
1.831
1.740
CS-6.03c, BS-3.46c
CS-4.82c, BS-3.79c
4.51 [16], this work
3.52 [9]
CS-4.88c, BS-2.87c
CS-4.12, BS-1.79
3.83
3.43 [13]
CS-3.45, BS-2.48
1.817
OS-2.82
1.803
1.799b
1.740
1.745
1.747
1.740
CS-4.02, BS-2.58
[Cr2{ArNCHNAr}2] Ar = 2,4,6-Me3C6H2 [4]
[Cr2{ArNCHNAr}2] Ar = 2,6-Et2C6H3 [4]
[Cr2{ArNCHNAr}2] Ar = 2,6-iPr2C6H3 [4]
[Cr2{ArNCMeNAr}2] Ar = 2,6-iPr2C6H3 [4]
a
b
c
CS-4.49, BS-2.79
Model species used in the quantum computations.
Optimized Cr–Cr distances obtained from CASPT2 calculations.
DFT BO’s computed at CASPT2 optimized CrCr distances.
complexes are characterized by the extremely short metal–metal
bonds of 1.74 Å, which were stated to be independent of the steric
hindrance of the ligands [4].
3. A historical overview of the theoretical methods
In this short prospective we have no aim to provide a complete
overview of all the theoretical efforts that have been devoted to the
understanding of the bonding in Cr–Cr complexes. Several excellent reviews exist on this topic [21,27,32–35] and the reader is invited to consult these Letters as well as the references therein.
Instead, we now present an account of the important steps in the
theoretical development of the description of the multiply bonded
Cr–Cr species over the last three decades.
The first attempts to address theoretically the multiple metal–
metal bond in the chromium diatom are due to the work of
Klotzbucher and Ozin [36] who used the extended Hückel (EH)
and SCF-Xa-SW molecular orbital techniques to investigate the
spectroscopic data and bonding in Cr2 obtained experimentally at
low-temperatures in an Argon matrix. As pertinently noticed by
Politzer et al. [27] in his 1999 review discussing these early results,
‘ironically, but prophetically, one of the more successful treatments of Cr2 was one of the earliest, and used the semi-empirical
extended Hückel technique’.
The most successful description of the multiple Cr–Cr bonding
was achieved, however, using the multiconfigurational quantum
chemistry methods. As far as these multiconfigurational methods
are concerned, the chromium diatom is ‘by excellence’ the golden
case to be probed, since its electronic ground state is highly multiconfigurational in character, as demonstrated by the weight of the
closed-shell Hartree–Fock configuration in the total wave function,
which is only 45% at the equilibrium geometry [37]. First multireference study of the Cr2 diatomic comes from Goodgame and Goddard [22], who used the spin-optimized generalized valence bond
(GVB) method including inter-pair correlations and van der Waals
interactions with 6000 configurations. They described the ground
state of the Cr2 as an antiferromagnetic dimer with very low lying
electronic states and noticed an absence of multiple bonding between the metal centers. In subsequent Letters with the modified
generalized valence bond (MGVB) method [38,39] they have also
predicted the double minimum for the potential energy curve of
Cr2 with short bond at 1.61 Å and a longer bond at >3.0 Å. The long
bond was described as a bearing a single 4s–4s interaction with the
d shells antiferromagnetically coupled into a net singlet state,
whereas to the short bond form were assigned five covalent bonds
from the chromium d–d orbitals. One of the most extensive studies
concerning the Cr2 diatom was undertaken by Dachsel et al. [40]
who reported a MRACPF calculation that used more than a billion
configurations, yielding a bond distance of 1.72 Å and a largely
underestimated bond energy of only 25.1 kcal mol1 (experimental = 35.3 kcal mol1). Early work involving a CASPT2 treatment
to account for the large dynamic correlation effects resulted in a
very reasonable description of Cr2, showing the utility of the multiconfigurational approach for the study of such highly correlated
system [41–43]. After the removal of a number of intruder states
that have appeared in the wave function deteriorating severely
the result, a highly accurate dissociation energy of 35.7 kcal mol1
and a slightly overestimated Cr–Cr distance of 1.71 Å (exp. 1.68 Å)
were obtained. This work was further extended by Andersson, who
used a modified form of the zeroth-order Hamiltonian in the CASPT2 method providing a detailed description of a large number of
Cr2 excited states [44].
However, these early studies are afflicted by errors due to the
choice of the basis set and the treatment of relativistic effects. A
large BSSE correction to the binding energy was obtained with
the basis set used in the work of Roos and Andersson’s, even if
an attempt was made to enlarge the basis set in the 3s, 3p region.
Scalar relativistic effects were introduced only at the lowest level
using perturbation theory with a non-relativistic basis set. The
problem was finally resolved in a Letter by Roos published in
2003 [37], where accurate spectroscopic constants and bond
length, as well as bond energy were obtained. This work was done
with a basis set of the atomic natural orbital type (ANO) [45] using
the Douglas–Kroll Hamiltonian [46] to account for scalar relativistic effects. The semi-core electrons were included in the correlation
treatment (3s, 3p for first row transition metals). The problem with
intruder states was solved with the aid of a modified zeroth-order
Hamiltonian (the IPEA shift [47]) and an extended active space.
In parallel to more demanding multiconfigurational treatments,
other methods have also been employed to describe these systems.
The earliest attempts to describe the Cr–Cr bond using the HFS (or
Xa) theory, which is an approximation to the Kohn–Sham DFT formulation, gave Cr–Cr bond and force constants, which were in a
reasonably good agreement with experimental values, however
4
M. Brynda et al. / Chemical Physics Letters 471 (2009) 1–10
the bond energies suffered notoriously from large errors. In addition these calculations were unable to predict accurately the shelf
region of the potential. The calculations using the local spin-density approximation by Delley et al. [48] and later on by Baykara
et al. [49] yielded a potential curve for Cr2 with a single minimum
and a highly anharmonic shape. The calculations using a modified
non-local density functional theory reported by Ziegler, Tschinke
and Becke gave better results [50], with a Cr–Cr distance of
1.65 Å and slightly overestimated dissociation energy to
40.4 kcal mol1. However, as shown in Fig. 1 in the first paragraph,
the characteristic feature of the experimental potential energy
curve for the Cr2 is that it has a shoulder at longer Cr–Cr separation
around 3.0–4.0 Å. Such a shoulder (which arises from the interaction of the 4s orbitals and will be discussed in more detail later)
should also be reflected in the calculated potential energy curves.
In 1994 Bauschlicher and Partridge [32] calculated the potential
energy curves for Cr2 using three popular functionals, commonly
used for the calculations of the multiply bonded transition metal
complexes, namely pure BLYP and hybrid B3LYP and B3P86. In
his report he drew attention to the fact that only the BLYP curve
displayed a slight shoulder between 2.5 and 3.0 Å, in agreement
with experiment, unlike the analogous curves obtained with the
two other functionals. This work suggested that use of the hybrid
functionals (e.g. B3LYP, B3P86) is not appropriate in the case of
strongly correlated electrons and provides a very poor description
of the Cr2 potential energy curve. Several years later, Edgecombe
and Becke analyzed this issue by performing calculations on the
ground state potential energy curve of Cr2 using three different
DFT approaches: SVWN, BLYP, and B3P86. These calculations performed within the broken symmetry approach with the spin projection method proposed by Noodleman to correct the ground
state energy [51,52] used a large basis set (uncontracted Huzinaga
14s9p5d2f appended with two 4f functions). Very interestingly,
only the potential energy curve obtained with the hybrid B3P86
functional showed a double well with a first minimum at 1.59 Å
and a second one at 2.40 Å. The corresponding dissociation energies were calculated to be 31.8 and 26.3 kcal mol1 respectively
[53]. More recently, the DFT calculations have been also used to
probe the electronic structure of other Cr–Cr species. In 2001
Weinhold and Landis predicted a trans-bent structure for the quintuply bonded HCrCrH species [54], several years before the synthesis of the Ar0 CrCrAr0 was achieved.
After the report of the experimental results on the Ar0 CrCrAr0 , a
multitude of DFT studies were published, mainly with the aim of
characterizing the extent of multiple bonding in the experimentally reported Cr–Cr core, as well as to determine the geometrical
features in the synthetic Cr–Cr complexes. Yet, all of these reported
studies are based on the assumption that the DFT can be used in its
monodeterminental formulation to probe the Cr–Cr bonding by
treating the chromium–chromium interaction as a unique closedshell spin configuration. This assumption might be considered
legitimate, because of the largest apparent weight of this configuration in a CASSCF wave function. Such an approach gives, no
doubt, a very simplified picture of the bonding situation; however,
there is a qualitative agreement between the molecular orbitals involved in the Cr–Cr bonding obtained from multiconfigurational
methods and those obtained from DFT.
4. The theoretical approach
The discussion of the electronic structure and the chemical
bond in the molecules studied in this Letter will be based on multiconfigurational wave functions, in particular, the Complete Active
Space (CAS) SCF method [45], as well as Density Functional Theory
(DFT) [55]. The CASSCF method constructs a wave function as a full
CI in a subspace of the molecular orbitals, the active space. This
orbital space has to be chosen such that it reflects the multiconfigurational character of the wave function during the chemical process under study. In addition, a number of inactive orbitals will be
doubly occupied. They are assumed to be unaffected by the
changes in the electronic structures that may take place. The
choice of these orbital spaces is the key to a successful CASSCF
calculation.
The CASSCF wave function may be analyzed in terms of its natural orbitals and their occupation numbers, which describe different correlation effects. If the active space includes the important
orbitals, they will remain stable when the active space is increased,
ultimately to the full CI limit. They are therefore useful tools to
discuss the electronic structure of the studied molecules. In multiconfigurational quantum chemistry, a single chemical bond is
described by a pair of orbitals, a bonding and an antibonding
one. Usually, their occupation numbers add up to 2.0. Thus, two
electrons reside in these two orbitals. If the occupation number
of the bonding orbital, gb, is close to two and the corresponding
antibonding orbital has a small occupation number, gab, there is
a fully developed chemical bond with a bond order equal to one.
This is the situation for the chemical bonds in most normal molecules at their equilibrium geometry. However, in transition metal
compounds with multiple metal–metal bonds, one often sees occupation numbers gb, which are smaller than two. If the two occupation numbers gb and gab are both close to one, we have no chemical
bond, but two antiferromagnetically coupled unpaired electrons.
We can therefore define a quantity, the effective bond order,
EBO, which quantifies the formation of a chemical bond [16]. For
a single bond it is defined as
EBO ¼ ðgb gab Þ=2
The EBO will be close to one for a fully developed single bond
and close to zero when the bond is dissociated. In multiply bonded
systems, one has to add up the individual values for each pair of
bonding and antibonding orbitals to obtain the total EBO. At transition states on an energy surface, one often finds half broken
bonds where the EBO is somewhere between the two limits. This
also often happens in multiply bonded systems that include orbital
pairs with weak overlap, for example, the d orbitals along a metal–
metal bond. We shall encounter such examples below. The EBO
gives us a means of quantifying the bond order concept for accurate wave functions. It is meaningless for Hartree–Fock wave functions or for closed-shell DFT, but could be useful in a broken
symmetry UHF situation if a natural orbital analysis is carried out.
The CASSCF wave function is used to analyze the bonding in a
given complex. In order to obtain accurate energies, one has to
add an estimate of the dynamic correlation effects. This will be
done here using multiconfigurational second-order perturbation
theory, CASPT2 [42]. Details of the calculations will be given for
each of the systems to be discussed. All calculations have been performed using the MOLCAS program system [45].
The DFT calculations reported here were carried out using the
B3LYP hybrid approximation to the exact exchange-correlation energy functional as implemented in the GAUSSIAN 03 program [56].
Preparation of the guess wave functions for the broken symmetry
approach was performed using the AOMIX program [57] by Gorelsky
(see the Ref. [57] for details). The potential energy curves of Cr2 and
the parent PhCrCrPh molecules were obtained by single point calculations at increasing Cr–Cr separation between 1.3 and 5.0 Å. The
basis set used for the description of Cr2 was the contracted
(7s6p4d2f1g) cc-pVTZ that accounts for the correlation of the
4s3d electrons [58]; the basis set used for the description of the
PhCrCrPh was the (5s3p2d) DGauss DGDZVP [59,60] internally
stored in GAUSSIAN 03.
M. Brynda et al. / Chemical Physics Letters 471 (2009) 1–10
The broken symmetry (BS) approach for the calculation of the
electronic structure of the transition-metal diatoms due to Noodleman’s work [51,52] has been discussed in several Letters [61,62],
and here we report only the most important elements of this DFT
ansatz. The approach for the transition-metal diatoms is based
on the Dirac-van Vleck Hamiltonian, which accounts for the exchange-coupling between the two metal centers:
^ HDmV ¼ 2J ^SA ^SB
H
Here J stands for the exchange-coupling parameter and ^SA and ^SB
define the local spin operators for centres A and B, respectively. In
case of DFT, the BS energy is obtained by a standard SCF procedure
using an unrestricted calculation with the appropriate guess wave
function, which leads to the BS solution with different orbitals for
different spin. To calculate J it is also necessary to compute the
high-spin (HS) state with MS = SA + SB = (Smax). With these quantities
in hand, the J can be evaluated based on Noodleman’s equation as:
J¼
EHS EBS
S2max
An alternative approach was proposed by Yamaguchi et al. [63,64],
where J is computed as:
J¼
EHS EBS
^
2
hS iHS h^S2 iBS
Note that the Yamaguchi equation is valid for the entire range of
coupling strengths and reduces to the Noodleman equation only
in the weak coupling limit [61]. The energy of the ground state is
then obtained using the following equation:
EGS ¼ EBS þ Smax J
In case of the Cr–Cr species, the 3d electrons are assumed to be antiferromagnetically coupled and SA = SB = 5/2. Therefore, Smax = 5 and
the ground state energy is obtained as:
EGS ¼ EBS þ 5J
For all DFT calculations reported in the recent Letters on Cr–Cr
species and discussed in the following paragraphs, the level of theory used by the authors of the original experimental or theoretical
Letters is described in the corresponding references and it is omitted here.
The bond orders (BO) calculated with DFT using AOMIX [57] are
represented by Wiberg bond indices defined as
BAB ¼
XX
ðPSÞba ðPSÞab
5
quantify theoretically because of the strong correlation effects that
occur due to the crowdedness of electrons in the bonding region.
CASPT2 calculations [37] have, however, yielded spectroscopic
constants and a potential curve in good agreement with experiment. We can thus conclude that its electronic structure is also
well described at this level of theory. In Fig. 2 are shown the most
important active orbitals obtained from the CASPT2 calculation.
There are six bonding orbitals (4srg, 3drg, 3dpu, and 3ddg) and
six corresponding antibonding orbitals (4sru, 3dru, 3dpg, 3ddu).
Bonding and antibonding orbitals are paired together. One notices
that in each pair the two occupation numbers add up to 2.00,
which makes it possible to attach an EBO to each pair. The first pair
is formed from the 4s atomic orbitals forming a r bond with an
EBO of 0.90. It has been argued that this orbital is more to be regarded as a Rydberg orbital (see for example Ref. [21]) than as a
truly bonding orbital. It is true that at the equilibrium bond distance (1.68 Å), the 4s–4s interaction is repulsive because of the
much larger size of the 4s orbitals compared to the 3d’s. The 4srg
orbital is, however, doubly occupied with only a small occupation
of the corresponding antibonding orbital 4sru. These two orbitals
will not play any role in the description of the Cr–Cr multiple bond
in the complexes to be described below. The two electrons will
move to the ligands, leaving a Cr–Cr moiety with at most five electron pairs.
These five pairs form the 3d bonds in the Cr–Cr complexes. As
can be seen in the figure, their binding power differs substantially.
The 3dr-orbital pair makes a bond with an EBO of 0.77, while the
3dp bonds exhibit an EBO of 0.81 each, and the 3dd bonds an EBO
of 0.58 each. The total EBO is therefore 4.45. So, is the bond in the
Cr–Cr diatom a sextuple, quintuple, or even only a quadruple
bond? The last conclusion could be derived from the fact the d
bonds are very weak and could also be considered as two pairs
of antiferromagnetically coupled 3dd atomic orbitals. This observation will be useful in the discussion of the bonding in some of the
complexes presented later, keeping in mind that these orbitals can
instead form bonds with the ligands.
The CASSCF/CASPT2 results presented above [37] yield a bond
distance of 1.66 (1.68) Å and a bond energy (D0) of 38.0
(35.3) kcal mol1 where the experimental values are given within
parentheses. The potential curve is shown in Fig. 5 together with
the experimental curve from the work of Casey and Leopold [21]
and the corresponding curves obtained from DFT calculations that
will be described in the DFT section.
The bonding situation described above (with, as noticed, the
exception of the 4s electrons) is also valid for the analogous
a2A b2B
where P is the total density matrix and S is the overlap integral
matrix.
5. The chromium diatom and other Cr–Cr species studied with
the CASSCF/CASPT2 method
We shall start this exploration of the chromium–chromium
multiple bond with a discussion of the bond between two Cr(0)
atoms, the Cr2 diatom. Very recently, this diatom has been used
to illustrate the ability of the DFT method to compute potential
curves for multiply bonded systems [65]. Below we shall perform
a comparison between results obtained with the CASPT2 and DFT
but first we shall take closer look at its electronic structure. This
will be helpful when we try to understand the Cr–Cr bond in more
complex systems.
The Cr–Cr bond is formed from the six unpaired electrons on
each of the atoms that have the ground state (3d)5(4s)1, 7S. Thus,
in principle, a sextuple bond can be formed between the two chromium atoms. It is not surprising that such a bond is difficult to
Fig. 2. The natural orbitals for the chromium diatom. Orbital labels and occupation
numbers are given below each orbital (contour lines at the density 0.07 e/au3).
6
M. Brynda et al. / Chemical Physics Letters 471 (2009) 1–10
Cr–Cr species, where the chromium atoms are attached to the ipsocarbon (the ring carbon atom bearing the substituent at position 1)
of a neighboring aryl ligand. Such species are represented by the
room-temperature stable Ar0 CrCrAr0 complex (cf. Fig. 3) as well
as its synthetic analogues with modified substituents on the terphenyl ligand.
The Cr–Cr bonding in Ar0 CrCrAr0 was first analyzed using a simplified model [9] where the ligands were replaced by two phenyl
rings, the PhCrCrPh. The interaction of the two chromium atoms
leads in this case to five, rather than six, metal–metal bonding
molecular orbitals, along with their antibonding counterparts.
The CASPT2 optimization results in two structures. The first one
is trans-bent, with the computed Cr–Cr and Cr–C distances of Cr–
Cr = 1.75 Å and Cr–C = 2.02 Å respectively. Note that these distances are slightly shorter than the experimentally determined values in the case of the corresponding Ar0 CrCrAr0 (Cr–Cr = 1.83 Å, Cr–
C = 2.15 Å) and this discrepancy is believed to arise from additional
weak interactions with the ligands. Such an interaction is attributed to the extra aryl substituents in Ar0 CrCrAr0 , which because
of steric and electronic factors as well as the presence of an additional Cr–Phenyl ring interaction weaken both the Cr–Cr and Cr–
C bonds somewhat. The second structure is linear, with a short
Cr–Cr bond of 1.678 Å and Cr–C bond of 2.040 Å. It lies only
1 kcal mol1 lower in energy than the trans-bent structure. The
Cr–Cr bond energy, obtained by comparing the energy of the CAS-
Fig. 3. Molecular structure of the model species for the Ar0 CrCrAr0 recently
synthesized by Power’s group [3], where isopropyl groups were replaced with
methyls.
Fig. 4. Molecular structure of model species for the Cr2 diazene recently synthesized by Kreisel et al. [5], where the isopropyl groups were replaced with methyls.
Fig. 5. The potential energy curve for the chromium diatom obtained with the DFT
calculations using a B3LYP functional and an acc-pTZVP basis set (see text for
details). The CASPT2 curve (black solid line) and the experimental curve [21] (black
dashed line) are provided for comparison. HS – high-spin curve (S = 5), BS
uncorrected – broken symmetry open singlet curve without energy correction; BS
Noodleman corrected – broken symmetry open singlet curve with energy correction according to Noodleman’s equation; BS Yamagushi corrected – broken
symmetry open singlet curve with energy correction according to Yamagushi’s
equation.
PT2 optimized PhCrCrPh complex with that of two CrPh units (with
DFT-optimized geometry), yields a value of 76 kcal mol1. This is
almost twice the bond energy found for Cr2 (36 kcal mol1). As discussed in the case of Cr2, the interaction of the 4s electrons is repulsive at equilibrium geometry, causing an important weakening of
the Cr–Cr bond. Such weakening does not occur in the RCrCrR
structures because of the direct involvement of the 4s electrons
in the Cr–C bond with the ligand (see however the discussion below on possible remaining effect of the 4s orbital on the Cr–Cr bond
distance).
The analysis of the corresponding CASSCF wave function shows
that the major configuration has a total weight of 45% and all the
bonding orbitals occupied (rg)2(pu)4(dg)4. The second dominating
configuration has a weight of 9% and corresponds to a double excitation (dg)2 ? (du)2 showing the weakness of the 3dd bond. The
analysis of the occupation numbers in a manner analogous to the
Cr2 case, shows the following: The computed values for the Cr–Cr
bonding orbitals and their antibonding counterparts in PhCrCrPh
are rg(1.79), ru(0.21), pu(1.76), pg(0.24), pu(1.79), pg(0.21),
dg(1.69), du(0.31), dg(1.50), du(0.50), yielding a total EBO of 3.52.
The fractional EBO’s for r, p and d bonds are r = 0.79, two p bonds
of 0.76, 0.79 and two d bonds of 0.69 and 0.50. Thus we see that the
fractional EBO’s for the CrCr bond are almost identical with the ones
obtained for the Cr2 and the difference between the total EBO’s calculated for these species (3.52 vs. 4.45) is due only to the remaining
single bond (with fractional EBO of 0.90) formed by the 4s electrons
in the additional Cr–C bond with the ligand. Finally we can now
have a quick look at the entire Ar0 CrCrAr0 species where the ligands
were kept to include the secondary interaction of the chromium
atoms with the p system of the ligand [13]. The total EBO is 3.43,
in fact almost identical to the one obtained for PhCrCrPh [9].
When the terpehenyl moieties are replaced with the ligands containing nitrogen such as the diimines, the situation changes. Even if
the total EBO is similar in the case of the RCrCrR and RN2CrCrN2R
species, the bonding situation and the interactions involved are
somewhat different. We have recently described such a bonding in
a model compound [12] based on the X-ray structure of the
corresponding experimental species reported by Kreisel et al. [5]
(see Fig. 4).
M. Brynda et al. / Chemical Physics Letters 471 (2009) 1–10
7
Fig. 6. The 12 CASSCF natural orbitals with their occupation numbers in the Cr2(diazadiene)2 complex. Orbitals 9–12 shows the combined Cr–Cr d bond and Cr–N p bond. The
contour line used is 0.05 e/au3.
The leading configuration in the CASSCF wave function is, in a
manner, identical to all the other compounds studied here, since
it is the closed-shell configuration (Cr–Cr)r2(Cr–Cr)p4(Cr–
Cr)d2(Cr–N)p4, with a weight of 60%. It is interesting to notice that
all the other configurations contribute to the wave function with
weights lower than 5%. Since additional atoms (nitrogens) are
present in the ligands, one has to include these atoms in the active
space. By using 12 electrons in 13 orbitals we have obtained 10
molecular orbitals, which are almost completely localized on the
Cr–Cr moiety, and two more delocalized orbitals which extend to
the nitrogen atoms through bonds to the imine p orbitals. The
presence of such delocalized orbitals in the active space makes
the evaluation of the EBO less straightforward, for the evident reason that their contribution to the Cr–Cr bond cannot be directly assigned. It is, however, possible to carry out a localization of the
orbitals such that the orbitals located at the Cr–Cr moiety are
clearly defined. These orbitals are no longer eigenfunctions of the
density matrix, but the effect of the off-diagonal elements is small
and may be neglected in the calculation of the EBO [12]. Such a
procedure was applied and yielded a total EBO of 3.43. The fractional EBO’s obtained from the localized orbitals, which can now
be simply assigned as being involved in the CrCr bonding only
without any ambiguity, are quite interesting. The occupation numbers for the Cr–Cr bonding orbitals and their antibonding counterparts are rg(1.81), ru(0.17), pu(1.81), pg(0.18), pu(1.80), pg(0.19),
dg(1.70), du(0.29), dg(0.75), du(0.19), yielding a total EBO of 3.43.
The fractional EBO’s are again very similar to the other compounds
for all the orbitals, except for the d interaction, for which the occupation numbers are much smaller. This is due to the fact that this
localized orbital was obtained through a localization procedure
from the orbital, in which initially an important part of the bonding
occurred with the neighboring nitrogen atoms in the form of a Crimine p bond [12] (cf. Fig. 6).
6. The chromium diatom and other Cr–Cr species studied with
DFT
Since we have already discussed in details the bonding in Cr2
and in several larger molecules containing the Cr–Cr core in the
multiconfigurational methods section, here we would rather present shortly the analogous DFT results and later on, discuss the sim-
ilarity and the differences between the two theoretical
descriptions. The DFT in its monodeterminental formulation,
which is commonly used by most of the inorganic chemists and
more generally by the experimentalists, can only describe the
bonding in a Cr–Cr moiety either as a closed-shell system, or as
an open shell determinant with MS = 0, in which all or a fraction
of the unpaired electrons on both metal centers are antiferromagnetically coupled. The DFT ‘wave function’ is in the latter case not
an eigenfunction of spin and symmetry and is commonly referred
to as a broken symmetry (BS) wave function. The spin symmetry
can be approximately restored using the Noodleman spin Hamiltonian, as described in the theoretical section.
The present calculations were performed with the hybrid functional B3LYP, because of its widespread use in the majority of the
DFT studies describing the above mentioned experimental results.
It is, no doubt, possible to look into the same problem with other
functionals but this is not the point here. It is important to recall
that within the B3LYP approximation to the exact exchange-correlation functional, the Cr2 (as well as PhCrCrPh or in general RCrCrR
species) remain unbound if computed in the closed-shell singlet
state (CS). If we compare now the potential curves for the Cr2 molecule (Fig. 5) with the analogous potential curves obtained with
the CASPT2 method, several things are striking. First, let consider
the closed-shell potential. It is not surprising that the DFT description of the closed-singlet state is not appropriate in describing the
experimental curve for Cr2. The energy is too high and the obtained
minimum corresponds to a Cr–Cr bond length of 1.56 Å, 0.1 Å
smaller than experiment. It is worth noting that such an underestimation of the Cr–Cr bond distance arises with most of the popular
DFT functionals not only in the optimized structures of Cr2 but also
of the analogous RCrCrR species [3,9,13].
Let us now switch to the high-spin curve. As expected, due to
the energetically unfavorable ferromagnetic coupling between
the electrons on both chromium centers with two MS = 5/2 (note
that the total spin is S = 5 and the multiplicity 2S + 1 = 11), the
curve presents a minimum only at a very large separation of
2.8 Å, which corresponds to the 4s–4s bond. The curve obtained
with the uncorrected low spin BS solution alone gives a minimum
at about 2.5 Å. However, as can be seen from the Fig. 5, the spin
projection technique performed either via the Noodleman or the
Yamagushi correction results immediately in a better agreement
8
M. Brynda et al. / Chemical Physics Letters 471 (2009) 1–10
with the experimental parameters, yielding minima at 1.73 Å and
1.80 Å, respectively. In addition, the shoulder on the curve that
arises at about 2.30 Å is also reproduced. This is in line with the
previous finding by Becke, who obtained the spin projected broken
symmetry solutions for Cr2 diatom with a very similar shape [53].
The shape of the curves depends of course on the functional and
the basis set used, and one has to be careful in selecting the proper
basis set. Here we have used a large basis set with two f and one g
functions. The presence of the f functions has been reported to be
very important in the proper description of the strongly correlated
systems such as the multiply bonded chromium diatom [66–68].
However the use of a smaller basis set (DGDZVP, data not shown)
fortuitously results in a slightly better agreement for the equilibrium Cr–Cr bond distance (1.69 Å).
The dissociation energy of 32.7 kcal mol1 calculated at the
equilibrium distance as the difference between the energy the
two Cr fragments minus the energy of the Cr2 molecule (with Noodleman’s energy correction) is slightly lower than the experimental one (36.0 kcal mol1), and also as compared to 38.0 kcal mol1
obtained from the CASPT2 calculations [37].
Analogous calculations were also performed on the PhCrCrPh
model (data not shown). In this case, the dissociation curve as obtained from DFT is similar to the one obtained for Cr2. The major
difference is the equilibrium distance of 1.70 Å (as compared to
1.75 obtained from the CASPT2 optimization). A characteristic feature of the spin-projected BS curve is also the absence of the shoulder in the vicinity of 2.5 Å, which was observed in the spinprojected BS curve of Cr2. This is easily explained by the absence
of the 4s electrons, which are now being involved in the Cr–C bond
with the phenyl rings. The calculated dissociation energy
(36 kcal mol1) is however much lower than the energy obtained
from the CASPT2 calculations
After this more detailed description of the two model species,
we shall discuss other DFT studies, which concern synthetic Cr–
Cr compounds. The already mentioned Ar0 CrCrAr0 was recently
investigated with DFT within a closed-shell description. The DFT
computed bond order was 4.12 and the molecular orbitals obtained from the closed-shell description are similar to the ones obtained form the CASSCF wave function [3,13]. An additional
difficulty in this compound is the secondary interaction of chromium with the flanking aryl, which is the origin of the slight elongation of the Cr–Cr bond length as compared to the PhCrCrPh
model. DFT is useful in analyzing such a secondary interaction. Unlike the chromium–chromium bonding that cannot be analyzed
properly with the closed-shell DFT, in the case of the secondary
interactions of the metal with the ligands, DFT performs relatively
well. Our calculations on both restrained models for ArCrCrAr and
on the simplified Cr–Benzene species indicate that the interaction
between the Cr atom and the arene system is weak (as compared
to much stronger interaction that occurs for example in the case
of the corresponding Fe and Co species [13]).
The DFT closed-shell studies performed on the Cr2(diazadiene)2
complex by Kreisel et al. [5] result in reasonable agreement of the
optimized geometries with the geometrical features of the experimental X-ray structure. There are six molecular orbitals involved
in the bonding, of which five are localized on the Cr–Cr fragment
and the additional one is a p type Cr–N interaction. This is also consistent with the MO picture obtained from the CASPT2 study, however the DFT computed BO is as usual too high (4.28) as compared
to the one obtained from the CASSCF wave function (3.43).
Since we have been discussing the differences between the DFT
and CASSCF pictures of the bonding in Cr–Cr species, an important
problem that arises when the computed bond orders are compared between the two approaches should be briefly discussed.
Inspection of Table 1, shows that the computed bond orders at
experimental Cr–Cr distances using DFT in both, the closed-shell
and broken symmetry approach are different from the EBO’s computed with the CASPT2. Clearly, the closed-shell DFT calculations
always overestimate the Cr–Cr bond order, while in the broken
symmetry approach, the inverse effect is observed. For example,
for the PhCrCrPh at the equilibrium distance the CASPT2 computed EBO is 3.5 while the DFT calculations result in a BO of 4.2
for the CS state and 2.8 for the BS state. Quite surprisingly, the
CASPT2 computed EBO is in this case the average of the BO values
computed for CS and BS states. Similar behavior is observed for
the other species including the complexes with the nitrogen containing ligands.
7. Discussion
The ultimate goal of the theoretical studies focused on the multiple metal–metal bonds is to understand the bonding pattern in
terms of the corresponding electronic structure. Such understanding can help us to answer several questions, e.g. why is one bond
shorter than the other, or how the bond order is related to the bond
length. As far as the multiple Cr–Cr bond is concerned, several topics are of interest. The first one is the effective bond order of the
Cr–Cr bond. It is important to understand how it is influenced by
the ligands, or more precisely, how the electronic structure of the
ligand affects the bonding between the chromium atoms.
How long is therefore the ‘natural’ bond length for the quintuple
Cr(I)–Cr(I) interaction? We have seen that the bond length in Cr2 is
1.68 Å and one might think that this is a lower limit, considering the
formation of a sextuple bond in this case. However, we also know
that the 4s–4s interaction is repulsive at this distance. What happens if we remove the 4s electrons? This can be done by adding
an electronegative ligand to each Cr atom, as for example in the
hypothetical compound FCrCrF. Optimization of the geometry of
this molecule at the CASPT2 level of theory with ANO-RCC-VTZP basis sets gives a Cr–Cr distance of 1.65 Å, 0.03 Å shorter than in the Cr
diatom. The 4s electrons are effectively removed by the electronegative ligand and we may thus consider 1.65 Å as the ‘unbiased’
Cr(I)–Cr(I) bond length. This may sound somewhat surprising, considering that all the measured (and computed) Cr–Cr distances are
longer than 1.70 Å. The only exception so far is the value computed
for the linear PhCrCrPh, which is 1.68 Å. The difference compared to
FCrCrF can be attributed to the more covalent bond formed between Cr and the phenyl group. At the trans-bent equilibrium
geometry of this system however, the computed bond length is
1.75 Å, causing a lengthening of 0.07 Å (the experimental value
for the Ar0 CrCrAr0 complex is 1.83 Å, but this difference, as we mentioned, can be attributed to the extra Cr–Aryl interaction [13]). It
seems that the bond length depends mainly on two factors, the
effectiveness in the removal of the 4s electrons and the involvement of the 3d orbitals in the Cr–L bonding. In trans-bent PhCrCrPh
this second factor is certainly important.
As demonstrated by the experimental findings and the CASPT2
calculations, the Cr–Cr bond seems to be fairly insensitive to the
modification of the ligands, even if different classes of ligands are
compared. For all the complexes that we have studied, with the
fivefold Cr–Cr interaction, the EBO ranges between 3.3 and 3.8. This
range is much narrower if only one class of ligands is taken into account (e.g. for the terphenyl ligands the EBO’s are 3.4–3.5). The second observation is that there is no simple relation between the Cr–
Cr bond length and the bond order. Moreover, there even seems to
be no simple relation between the type of the ligand and the same
bond length.
From a purely qualitative point of view, one would expect a
smaller bond order for a longer bond, based on the intuitive notion
of the overlapping molecular orbitals between the metal atoms.
However many different other factors can influence the bond
M. Brynda et al. / Chemical Physics Letters 471 (2009) 1–10
length. For example, in the case of the diimine ligands, the strain in
the NCCN ligand moieties is most probably the key factor that
holds the two Cr atoms in place and severely limits the variations
in the Cr–Cr bond distance.
This brings us to an interesting property that relates to the (relatively moderate) strength of the Cr–Cr quintuple bond, which we
have recently described as its apparent ‘robustness’. In a recent theoretical Letter we have addressed this Cr–Cr distinctiveness [13],
which, while not relevant to the di-chromium molecule, becomes
very important in the case of the bulky ligands used for the stabilization of the Cr–Cr bond, namely the secondary interactions with
the arene part of the ligand. Such interaction could, in principle,
be responsible for large variations in the Cr–Cr bond length. However, recent reports of the species where Cr–Cr core is attached to
very different ligands, where such interaction is absent, seems to
contradict this hypothesis. On the other hand, dramatic differences
between species where the atoms forming the metal–metal cores
are different from chromium seem to elucidate this problem. Recently, we have demonstrated that, for example, the Co–Co and
Fe–Fe dimers synthesized with the same ligand as the ones used
in Ar0 CrCrAr0 are not formed. This is because the metal ions themselves interact strongly with the arene parts of the ligands, precluding a significant multiple bonding between the metal centers.
Instead ‘half-metallocenes’ are obtained with the metal ions complexed in the 6g fashion to the arene moiety of the ligand. The
DFT calculations performed on simplified molecular models to
probe the extent of this arene–metal interaction, suggest that at
least two factors contribute to the structural differences observed
between the quintuply bonded chromium dimer and its cobalt
and iron congeners. Our conclusion is that the robustness of the
Cr–Cr quintuple bond is related to the special character of the RCr(I)
species, whose electronic (d5) structure seems to preclude strong
interactions with the nearby arenes, such as benzene or phenyl
fragments. This is most probably due to the fact that Cr(I) is particularly stable due to the half-filled d-shell, which results in a large
number of favorable intra-atomic exchange interactions that reduce Coulomb repulsion. Therefore Cr(I) is reluctant to lose these
favorable interactions, which greatly diminish its affinity to interact
with other ligands. This apparent reluctance to interact more
strongly with the surrounding p system of the ligand is an important factor that contributes to the stability of the quintuple Cr–Cr
bond. The second factor is related to the most important attribute
of the Cr(I)–Cr(I) bond, which is the presence of a high number
(10) of valence electrons that exactly match the number of bonding
molecular orbitals. The additional experimental evidence comes
from the fact that the monomeric Ar–Co and Ar–Fe congeners of
monomeric Ar–Cr form easily direct adducts with the benzene molecule, while in the analogous monomeric Ar–Cr species the Cr atom
is complexed rather by PMe3 or THF molecules [69].
Another interesting aspect of the chromium–chromium interactions studied here is the oxidation state of the chromium atoms in
different Cr–Cr bonds. Is this oxidation state always the same? In
the simplest case of the RCrCrR species, the answer is trivial. The
oxidation state of Cr is +1. However, when the N-containing ligands
enter the game, the determination of the formal oxidation state is
not straightforward anymore. First, the diimine ligands can be considered either as entities with 2, 1, or even 0 charge. Such a redox ambiguity is typical of diimines, where several resonance
structures can in principle be considered. In the case of the Cr2(diazadiene)2 compound reported by Kresiel et al. [5], the bond lengths
in the ligand suggest a reduced form, e.g. an oxidation state between 1 and 2. Therefore the oxidation state of the Cr ions
should be between +1 and +2. We have argued that Cr atoms lose
the 4s electrons, which in this case would participate in the bond
of Cr atom with the ligands. In addition, p bonds are formed between Cr and N, half a bond per Cr–N pair. This gives a formal oxi-
9
dation number of +1.5. The situation is even more confusing with
the ‘lantern’ compound reported by Tsai et al. [7]. Here the three
diimine ligands forming the cage should have a total charge of
3, bringing the formal oxidation state to somewhere between
1.5 and 2.0 and this should result in a formation of a quadruple
bond. Still, the bond distance is shorter, 1.74 Å.
In order to test if we could also obtain an unbiased bond length
for the Cr(II)–Cr(II) interaction, we also performed CASPT2 calculations on F2CrCrF2. The optimization of this species did, however,
not lead to any bound species. Our conclusion is that two Cr(II) ions
will not form a bond due to the increased Coulomb repulsion and
the decreased bond order as compared to Cr(I). Instead, the short
bond lengths observed in the synthesized Cr(II)–Cr(II) compounds
have to be explained by the nature of the supporting ligands that
are always bi-dentate and hold the Cr(II) ions in place. Some examples of such compounds were given above.
8. Conclusions and perspectives
The present study has analyzed the Cr–Cr multiple bond as it
occurs in Cr2, as well as in different recently synthesized low-valent Cr–Cr complexes. This multiple bond is characterized by one
feature that is invariant and does not change with the nature of
the ligand or with the oxidation state of the Cr atom: that is, the
(3dr)2(3dp)4 triple bond. This bond is fully formed in all the species studied here and exhibits always an EBO close to 2.3. The
remaining d bonds are however much weaker. Depending on the
ligand attached to the metal center, a competition between the
Cr–Cr bonding and the Cr-ligand bonding involving the orbitals
participating in the d bonds occurs. For example in the case of
the imine ligands, Cr–N p bonds are formed instead of Cr–Cr d
bonds. Formation of Cr–N p bonds in turn lowers the bond order
and should in principle also lead to longer bond lengths. However
this is not always the case, due to the steric influence of the dentate
ligands, which lead to bond lengths that maximizes the Cr ligand
interaction at the expense of the small changes in the Cr–Cr bond
energy. We can therefore conclude that the weakness of the d
bonds is the most important factor that makes the Cr–Cr bond
length flexible and strongly dependent on the nature of the ligands.
The molecules discussed here have also been studied using DFT
with the B3LYP functional. This approach has several shortcomings
for the wave functions with such a strong multiconfigurational
nature as those describing the Cr–Cr multiple bond. Normally,
the closed-shell DFT wave function is not stable but degrades to
a broken symmetry solution if allowed to do so. The closed-shell
solution can be then useful for the optimization of the general
structural features of the low-valent Cr–Cr complex, but the Cr–
Cr bond length is underestimated by about 0.1 Å and need to be
re-optimized at a higher level of theory. The broken symmetry
solution is even less useful. It gives much too long Cr–Cr bond distances and the electronic structure is severely distorted with the S2
values different from zero. Energy wise, it is possible to correct
approximately for these shortcomings in the broken symmetry approach using the Noodleman or Yamaguchi techniques, as described above, but this approach will not correct other properties
and is difficult to use, for example, for the commonly performed
tasks such as the geometry optimizations.
The final conclusion is that in order to describe properly the electronic structure of the compounds studied here, we need the multiconfigurational wave function approach. This approach can
today be applied for increasingly bulky systems, making it available
for the real chemical complexes that are studied experimentally.
And clearly, the number of experimental reports concerning multiply bonded species with large ligands is growing very fast. For such
large molecules, DFT can be useful as a helping hand because the
10
M. Brynda et al. / Chemical Physics Letters 471 (2009) 1–10
CASSCF/CASPT2 approach has no yet reached the stage where full
geometry optimizations for system with about 100 atoms are possible. A careful combination of DFT and the wave function approach
becomes therefore quite useful in many cases. However, this may
change in the near future due to the development of the Cholesky
decomposed gradient techniques [70] that will allow geometry
optimizations of large molecular systems at the CASSCF level.
Acknowledgments
M.B. thanks Prof. Philip P. Power for useful discussions, corrections and suggestions to the present manuscript. L.G thanks Swiss
National Science Foundation (Grant 200020-120007) for financial
support.
References
[1] F.A. Cotton, C.A. Murillo, R.A. Walton, Multiple Bonds between Metal Atoms, in:
F.A. Cotton, C.A. Murillo, R.A. Walton (Eds.), third ed., Springer, New York,
2005.
[2] F.A. Cotton et al., Science 145 (1964) 1305.
[3] T. Nguyen, A.D. Sutton, M. Brynda, J.C. Fettinger, G.J. Long, P.P. Power, Science
310 (2005) 844.
[4] C.-W. Hsu, J.-S.K. Yu, C.-H. Yen, G.-H. Lee, Y. Wang, Y.-C. Tsai, Angew. Chem.,
Int. Ed. 47 (2008) 1.
[5] K.A. Kreisel, G.P.A. Yap, O. Dmitrenko, C.R. Landis, K.H. Theopold, J. Am. Chem.
Soc. 129 (2007) 14162.
[6] A. Noor, F.R. Wagner, R. Kempe, Angew. Chem., Int. Ed. 47 (2008) 7246.
[7] Y.-C. Tsai, C.-W. Hsu, J.-S.K. Yu, G.-H. Lee, Y. Wang, T.-S. Kuo, Angew. Chem., Int.
Ed. 47 (2008) 7250.
[8] R. Wolf et al., Inorg. Chem. 46 (2007) 11277.
[9] M. Brynda, L. Gagliardi, P.-O. Widmark, P.P. Power, B.O. Roos, Angew. Chem.,
Int. Ed. 45 (2006) 3804.
[10] L. Gagliardi, G. La Macchia, M. Brynda, Abstracts of Papers, 236th ACS National
Meeting, Philadelphia, PA, United States, 2008.
[11] L. Gagliardi, B.O. Roos, Lect. Ser. Comput. Comput. Sci. 6 (2006) 6.
[12] G. La Macchia, F. Aquilante, V. Veryazov, B.O. Roos, L. Gagliardi, Inorg. Chem. 47
(2008) 11455.
[13] G. La Macchia, L. Gagliardi, P.P. Power, M. Brynda, J. Am. Chem. Soc. 130 (2008)
5104.
[14] C.R. Landis, F. Weinhold, J. Am. Chem. Soc. 128 (2006) 7335.
[15] G. Merino, K.J. Donald, J.S. D’Acchioli, R. Hoffmann, J. Am. Chem. Soc. 129
(2007) 15295.
[16] B.O. Roos, A.C. Borin, L. Gagliardi, Angew. Chem., Int. Ed. 46 (2007) 1469.
[17] A. Kant, B.H. Strauss, J. Chem. Phys. 45 (1966) 3161.
[18] E.P. Kündig, M. Moskovits, G.A. Ozin, Nature 254 (1975) 503.
[19] V.E. Bondybey, J.H. English, Chem. Phys. Lett. 94 (1983) 443.
[20] B. Simard, M.-A. Lebeault-Dorget, A. Marijnissen, J.J. ter Meulen, J. Chem. Phys.
108 (1998) 9668.
[21] S.M. Casey, D.G. Leopold, J. Phys. Chem. 97 (1993) 816.
[22] M.M. Goodgame, W.A. Goddard III, J. Phys. Chem. 85 (1981) 215.
[23] C.J. Barden, J.C. Rienstra-Kiracofe, H.F. Schaefer III, J. Chem. Phys. 113 (2000)
690.
[24] E.A. Boudreaux, E. Baxter, Int. J. Quantum Chem. 85 (2001) 509.
[25] E.A. Boudreaux, E. Baxter, Int. J. Quantum Chem. 100 (2004) 1170.
[26] E.A. Boudreaux, E. Baxter, Int. J. Quantum Chem. 102 (2005) 866.
[27] E.J. Thomas, J.S. Murray, C.J. O’Connor, P. Politzer, Theochem 487 (1999) 177.
[28] F.A. Cotton, S.A. Koch, M. Millar, Inorg. Chem. 17 (1978) 2084.
[29] S. Horvath, S.I. Gorelsky, S. Gambarotta, I. Korobkov, Angew. Chem., Int. Ed. 47
(2008) 1.
[30] E. Peligot, C.R. Acad. Sci. 19 (1844) 609.
[31] E. Peligot, Ann. Chim. Phys. 12 (1844) 528.
[32] C.W. Bauschlicher Jr., H. Partridge, Chem. Phys. Lett. 231 (1994) 277.
[33] S.R. Langhoff, C.W. Bauschlicher Jr., Annu. Rev. Phys. Chem. 39 (1988) 181.
[34] M.D. Morse, Chem. Rev. 86 (1986) 1049.
[35] D.R. Salahub, Adv. Chem. Phys. 69 (1987) 447.
[36] W. Klotzbuecher, G.A. Ozin, Inorg. Chem. 16 (1977) 984.
[37] B.O. Roos, Collect. Czech. Chem. Commun. 68 (2003) 265.
[38] M.M. Goodgame, W.A. Goddard III, Phys. Rev. Lett. 48 (1982) 135.
[39] M.M. Goodgame, W.A. Goddard III, Phys. Rev. Lett. 54 (1985) 661.
[40] H. Dachsel, R.J. Harrison, D.A. Dixon, J. Phys. Chem. A 103 (1999) 152.
[41] K. Andersson, P.A. Malmqvist, B.O. Roos, J. Chem. Phys. 96 (1992) 1218.
[42] K. Andersson, P.A. Malmqvist, B.O. Roos, A.J. Sadlej, K. Wolinski, J. Chem. Phys.
94 (1990) 5483.
[43] B.O. Roos, K. Andersson, M.P. Fülscher, P.A. Malmqvist, L. Serrano-Andrés, K.
Pierloot, M. Merchan, in: L. Prigogine, S.A. Rice (Eds.), Advances in Chemical
Physics: New Methods in Computational Quantum Mechanics, vol. XCIII, John
Wiley and Sons, New York, 1996, p. 219.
[44] K. Andersson, Chem. Phys. Lett. 237 (1995) 212.
[45] G. Karlstroem et al., Comput. Mater. Sci. 28 (2003) 222.
[46] N. Douglas, N.M. Kroll, Ann. Phys. 82 (1974) 89.
[47]
[48]
[49]
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
[66]
[67]
[68]
[69]
[70]
G. Ghigo, B.O. Roos, P.A. Malmqvist, Chem. Phys. Lett. 396 (2004) 142.
B. Delley, A.J. Freeman, D.E. Ellis, Phys. Rev. Lett. 50 (1983) 488.
N.A. Baykara, B.N. McMaster, D.R. Salahub, Mol. Phys. 52 (1984) 891.
T. Ziegler, V. Tschinke, A.D. Becke, Polyhedron 6 (1987) 685.
L. Noodleman, J. Chem. Phys. 74 (1981) 5737.
L. Noodleman, E.R. Davidson, Chem. Phys. 109 (1986) 131.
K.E. Edgecombe, A.D. Becke, Chem. Phys. Lett. 244 (1995) 427.
F. Weinhold, C.R. Landis, Chem. Edu.: Res. Pract. Europe 2 (2001) 91.
R.G. Parr, W. Yang, International Series of Monographs on Chemistry 16, in: R.
Breslow, J.B. Goodenough, J. Halpern, J.S. Rowlinson (Eds.), Oxford University
Press, New York, Clarendon Press, Oxford, 1989.
M.J. Frisch et al., Gaussian, Inc., Pittsburgh PA, 2003.
S.I. Gorelsky, S.I. Gorelsky, AOMIX: Program for Molecular Orbital Analysis,
http://www.sg-chem.net/, University of Ottawa, 2007.
N.B. Balabanov, K.A. Peterson, J. Chem. Phys. 123 (2005) 064107.
N. Godbout, D.R. Salahub, Can. J. Chem. 70 (1992) 560.
C. Sosa, J. Andzelm, B.C. Elkin, E. Wimmer, K.D. Dobbs, D.A. Dixon, J. Phys.
Chem. 96 (1992) 6630.
F. Neese, J. Phys. Chem. Solids 65 (2004) 781.
E. Ruiz, J. Cano, S. Alvarez, P. Alemany, J. Comput. Chem. 20 (1999) 1391.
T. Soda et al., Chem. Phys. Lett. 319 (2000) 223.
K. Yamaguchi, Y. Takahara, T. Fueno, in: V.H. Smith (Ed.), Applied Quantum
Chemistry, Reidel, Dordrecht, 1986, p. 155.
P.E.M. Siegbahn, M.R.A. Blomberg, Int. J. Quantum Chem., in press.
B.H. Botch, T.H. Dunning, J.F. Harrison, J. Chem. Phys. 75 (1981) 3466.
A.D. McLean, B. Liu, Chem. Phys. Lett. 101 (1983) 144.
S.P. Walch, C.W. Bauschlicher Jr., B.O. Roos, C.J. Nelin, Chem. Phys. Lett. 103
(1983) 175.
R. Wolf, M. Brynda, C. Ni, G.J. Long, P.P. Power, J. Am. Chem. Soc. 129 (2007)
6076.
F. Aquilante, R. Lindh, T.B. Pedersen, J. Chem. Phys. 129 (2008) 34106.
Marcin Brynda did his undergraduate and graduate
studies at the University of Geneva, where he also
obtained his PhD in Physical Chemistry in 2000. He was
appointed young lecturer in 2001 at the same University. In 2002 he obtained a Grant for advanced
researcher from Swiss NSF and moved to University of
California, Davis, where he worked as visiting lecturer
with R. David Britt in advanced EPR of biological systems. In 2006 he was appointed Faculty Associate Specialist at UC Davis. He provided theoretical description
of many new transition-metal and main group complexes in collaboration with the research group of Philip
Power and he is an author and coauthor of more than 50
scientific papers in international journals. He is presently a Research Associate at
the University of Geneva.
Laura Gagliardi completed her undergraduate and
graduate education at the University of Bologna, Italy.
She received her PhD in 1996. From 1997 to 1998 she
was a postdoctoral associate at Cambridge University,
UK. From 1999 to 2001she was a research associate at
the University of Bologna. From 2002 to 2005 she was
lecturer at the University of Palermo. In November 2005
she was appointed associate professor at the University
of Geneva. In January 2009 she will join the University
of Minnesota as Professor. She is the author of about 90
publications in international journals. Her scientific
interests concern the development and employment
study of quantum chemical methods for the study of
molecular systems containing heavy elements and the prediction of novel inorganic
species and chemical bonds.
Björn O. Roos received his PhD in 1967 at the Stockholm
University. He continued to work at this university until
1977 when he moved to the Chemical Center, Lund
University, where he became full professor in theoretical
chemistry in 1983. He retired on 2002 but continues to
work as emeritus. His main research interest is in the
development of methods in quantum chemistry and their
applications in chemistry. The emphasis has in later years
been on transition-metal and heavy element chemistry.